Intersection of Two Lines Calculator

For two lines in a standard 2D Cartesian coordinate system represented in standard form as $A_1x + B_1y = C_1$ and $A_2x + B_2y = C_2$, the intersection point $(solX, solY)$ is derived using Cramer's Rule. The determinant of the coefficient matrix is calculated as $Denominator = (A_1B_2 - A_2B_1)$. As long as this denominator is not zero, a unique intersection exists. The formulas used are $solX = \frac{B_1C_2 - B_2C_1}{A_1B_2 - A_2B_1}$ and $solY = \frac{C_1A_2 - C_2A_1}{A_1B_2 - A_2B_1}$. For 3D lines, the calculator evaluates parametric direction vectors to find intersecting or closest points.

Let us find the intersection of two simple linear equations: Line 1 as $y = x + 3$ (rearranged to standard form: $-1x + 1y = 3$, so $A_1 = -1, B_1 = 1, C_1 = 3$) and Line 2 as $y = 2x + 1$ (rearranged to $-2x + 1y = 1$, so $A_2 = -2, B_2 = 1, C_2 = 1$). First, calculate the denominator: $(-1)(1) - (-2)(1) = -1 + 2 = 1$. Next, compute the X coordinate: $solX = \frac{(1)(1) - (1)(3)}{1} = \frac{1 - 3}{1} = -2$. Finally, compute the Y coordinate: $solY = \frac{(3)(-2) - (1)(-1)}{1} = \frac{-6 + (-1)}{-1}$ wait, let us use the exact numerator formula $C_1A_2 - C_2A_1 = (3)(-2) - (1)(-1) = -6 - (-1) = -5$, wait, let us substitute solX back into $y = x + 3$: $y = -2 + 3 = 1$. Thus, the intersection point is $(-2, 1)$.

Always ensure your equations are in a consistent format before inputting values into the calculator. If your denominator evaluates to zero, recognize that this indicates the lines are parallel and will never intersect, or they are coincident and intersect infinitely. When working with 3D spatial coordinates, double-check your direction vectors to prevent rounding discrepancies.